Input Data (Bayesian estimation)

Estimation ML/Bayesian estimation
1

My question is related to Gali chapter 8 (implemented by prof. Pfeifer) and the accurate way to read data from a file.

On github, you mentioned that all variables are in log-deviations from the steady state, but interest rate which is mean 0.

After reading the guide on observable variables, I proceeded the following way:

  • GDP (real quarterly volumes) → log(gdp) → get cylical component using univariate-HP filter
    Read it from the file like: y_obs = yhat or y_obs = y_gap

  • Nominal interest rate (net annual percentages) → make it quarterly percentages ln(1 + rate/400) → get cyclical component using univariate-HP filter
    Read it from the file like: i_obs = i

  • Inflation Rate (net quarterly percentages) → get cyclical component using univariate-HP filter
    Read it from the file like: pi_obs = pi

Question 1: You mentioned that interest rate is mean 0, not in log-deviations from steady state. HP-filter also demeans the data and the cyclical component is mean 0. Nonetheless, shouldn’t there be a difference between the way you match output gap and how you match inflation / interest rate given one is mean 0 and the others in log-deviations?

Question 2: Given that all variables (but the abovementioned ones) are in log-deviations, is it appropriate to just detrend variables (log consumption per capita, log NX per capita) using univariate HP filter and consider the cyclical component as log-deviation from steady state and read it like c_obs = c & c_NX = NX plus some shock to avoid stochastic singularity?

Question 3: Is it ok to just take the log difference of variables such as nominal exchange and read it like er_obs = er - er(-1) + eps_er. I don’t understand why this is appropriate…I saw a lot of posts where people read variables like this, but I did not really understand why - er in the model is log deviation from steady state while er_obs is basically log difference of nominal rates → growth rate. How do these two match?

Thank you

3
  1. I don’t understand the question. A log deviation from steady state is mean 0.
  2. Yes, you can do that, because you simply take out your concept of a trend.
  3. A log deviation from steady state is \hat x_t= log(x_t)-log(\bar x). Thus, the steady state part drops out in the difference:
    \hat x_t-\hat x_{t-1}= log(x_t)-log(\bar x)-(log(x_{t-})-log(\bar x))=log(x_t)-(log(x_{t-})
  4. What do you mean with univariate HP-filter? The one-sided HP filter?
4

  1. On github, you mentioned nominal interest rate separately from the other variables (as being in mean 0 rather than log-levels). I thought I had to read it from a file in a different way: interest rate and natural output are not in log-levels, but rather mean 0
    I did not understand the meaning of that statement and I got confused a bit.

    i_obs = cyclical component [ ln(1 + rate/400) ] //so basically read it like this and I assume it is correct

    pi_obs = cyclical component [pi ] //pi was already net quarterly percentages so I straight got the cyclical component without any other processing

    My question was related to the treatment of these two variables and if I should have matched them differently as I did read them the same from the file

    i_obs = i
    pi_obs = pi
    Basically, matched them both exactly to model variables

  2. Ok, so basically, you can take the cyclical component of any variable and match it exactly to an existing variable if I understood correctly (obs_variable = model_variable + shock) where the observable variable is the cyclical component using a one-sided filter. Maybe I’ve missed in the guide, but is there a suggested way to tackle variables that can be negative such as NX and you cannot take the log of?

  3. Ok - basically obs_variable = model_variable - model_variable(-1) + shock, where I take the log difference of my observable

  4. Yes, one-sided HP filter (hpfilter1s from eviews more precisely)

Thank you

5

One follow-up question regarding your guide

If you take the log difference of your data (non-zero mean) and then demean it (now zero mean), do you match it like this?

XLS File:
obs_variable = log(obs_variabile_volume) - log(obs_variable_volume-1) => growth rates ~ 0.01 mean ~so around 1% mean which does not match the model variable which is mean 0

=> I demean it like: log(obs_variabile_volume) - log(obs_variable_volume-1) - average(log(obs_variabile_volume) - log(obs_variable_volume-1))

Dynare:
obs_variable = model_variable - model_variable(-1)

Or how do you specify you demeaned the data in log differences eXplictly when you read from the file? My guess you do not have to, as you match demeaned log-differences which is 0 mean which is already the variable in the model.

6

I think I solved it myself (hopefully).

If you use one-sided HP filter or you demean the data in log-difference is the same and in both situations, you can match data like:

obs_variable = model_variable + measurement_error

P.S: The only thing is…what if the data in log differences is mean 0… demeaning it seems meaningless.

7
  1. The statement is about the comparison to the book. Gali uses the log level of the interest rate, not the percentage deviation from steady state.
  2. I don’t get where a shock will come from in your equation unless you assume measurement error. Regarding net exports, a typical way is to use the net export share as an observable instead of log net exports.
  3. Yes, you would match demeaned first differences, where you use